Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion

Abstract

Abstract. Dzhaparidze and Spreij [5] showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. [2], where it is shown that the quadratic variation of the log-returns determines the hedging strategy.

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model

“…Recently, the convergence (3) is extended in [2] to some class of stochastic processes which contains nonsemimartingales in general. Let W = {W t } t∈[0,T ] be a standard Brownian motion, and B H = {B H t } t∈[0,T ] be a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1), independent of the Brownian motion W .…”

“…In [12], the case of semimartingales was studied. In [2], the randomized periodogram estimator was studied for the mixed Brownian-fractional Brownian model, and the weak consistency of the estimator was proved. This article investigates the asymptotic normality of the randomized periodogram estimator for the mixed Brownianfractional Brownian model.…”

Research on The Lookback Option Pricing of Disney Based on BSM Model

“…Ma (2021) investigated the behavior of the mixed fractional Brownian exponential population growth system [9]. Azmoodeh and Valkeila (2013) showed that a randomized diagram used to represent the quadratic variation which derived from a semi-martingale on Brownian motion could also be used for a special type of continuous stochastic process [10].…”